l061011dualityintrpWEB

l061011dualityintrpWEB - S.Grant ECON501 1.6 Duality,...

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S.Grant ECON501 (Ref: MWG 3.H, 2.F and 3.J) 3 closely related questions:- 1. Starting with an expenditure or indirect utility function can we work “backwards” to recover the underlying direct utility function that would have generated it? ( DUALITY THEORY ) 2. Given a function x ( p, w ) that is claimed to be an uncompensated demand function, is there some locally insatiable preference-maximizing consumer behind it? ( INTEGRABILITY PROBLEM ) 3. Viewing choice behavior as primitive, when is a set of observed choices of a consumer consistent with our preference-based model of choice? ( REVEALED PREFERENCE ) 1 S.Grant ECON501 1.6.1 Duality Theory Constructing a utility function from an expenditure function e ( p, u ) . De f ne V u ( p ) © x R L + | p.x e ( p, u ) ª De f ne V u \ p À 0 V u ( p ) © x R L + | p.x e ( p, u ) for all p À 0 ª De f ne u ( x ) max u { x V u } max u { p.x e ( p, u ) for all p À 0 } Theorem 1: u ( · ) is quasi-concave and continuous in x . Theorem 2: e ( p, u )=m in x p.x s.t. u ( x ) u .
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S.Grant ECON501 1.6.2 Integrability Problem Given x ( p, w ) does there exist a locally insatiable preference-maximizing consumer for whom x ( p, w
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l061011dualityintrpWEB - S.Grant ECON501 1.6 Duality,...

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